Our main objects of the study are the Lie algebras \(W_n \otimes B\)  and \((W_n \ltimes A_n) \otimes B\) . But due to some technical reasons (see the proof of Proposition REF ), we need to consider several copies of \(A_n\) , i.e., \(W_n \ltimes (S \otimes A_n)\)  where \(S\)  is
any finite dimensional abelian Lie algebra. Let us denote \(\mathfrak {L}_{S, n} : = W_n \ltimes (S \otimes A_n)\)  and \(\mathfrak {L}_{S,n}(B) := (W_n \ltimes (S \otimes A_n)) \otimes B\)  and \(\mathfrak {L}:= W_n \ltimes A_n\) , where \(B\)  is a finitely generated commutative associative unital algebra over \(.\) We begin the Section  with basic definitions. In Section  we recall the known results for \(W_n\)  and \(\mathfrak {L}\)  along with tensor field modules (also known as Larson-Shen modules, see [1]}, [2]}). In this section we completely classify irreducible uniformly bounded \(\mathfrak {L}_{S,n} (B)\) -modules.
Let \(V\)  be an irreducible uniformly bounded \(\mathfrak {L}_{S,n} (B)\) -module.
We first note that \(S\)  can be reduced to at most one dimensional in Proposition . So we can consider \(V\)  as an irreducible uniformly bounded \(\mathfrak {L}(B)\)  or \(W_n (B)\) -module. In Proposition REF , we prove that the action of \(A_n \otimes B\)  is associative on \(V\) . So by application of
[3]} Theorem 4.5, \(V\)  is a single point evaluation module.
In the rest of the section we attend the case where \(V\)  is an irreducible uniformly bounded \(W_n (B)\) -module. In this case also, using the celebrated technique of \(A_n\) -cover [4]}, we prove that \(V\)  is a single point evaluation module.
